Ethan D. Bloch

Proofs and fundamentals is a textbook for a one-semester course emphasizing proof-writing in the context of the core topics of sets, functions, relations, and cardinality. The target audience is "a wide variety of students" who have finished calculus and are moving on to more abstract mathematics. The table of contents does a good job of summarizing the subject matter.

Part I: PROOFS

Part II: FUNDAMENTALS

Part III: EXTRAS

1. Informal Logic

3. Sets

7. Selected Topics

2. Strategies for Proofs

4. Functions

8. Number Systems

5. Relations

9. Explorations

6. Infinite and Finite Sets

Parts I and II are designed to form the core of the one-semester course. It is intended that instructors then choose a few topics from Part III according to their taste.

Bloch's book is one of a growing genre of transition texts focused on rigor and basics. In the introduction, Bloch cites five texts similar to his own and I've seen quite a few more as well. I have pedagogical concerns about the entire genre. See what you think about the following two points in Bloch's text.

In Section 1.5, it is proved that certain hypotheses imply that there exists a stupid cat. The proof makes explicit reference to Existential Instantiation, Universal Instantiation, Modus Ponens, Modus Tollens, Modus Tollendo Ponens, De Morgan's Law, and Existential Generalization, among other things.

In Section 6.4, it is emphasized that we cannot be sure that the equations c1 = 4 and cn+1 = 3 + 2 cn together define a sequence. However Theorem 6.4.1 comes to our rescue. It says "Let b:R→R and h:R→R be given. Then there is a unique function f:N→R such that f(1) = b and that f(n+1) = h(f(n)) for all n∊N." The proof of this theorem involves a forward reference to Theorem 8.2.1 whose statement involves Henkin sets and whose proof occupies the two pages of Section 8.7.

My feeling is that this sort of material is not best for a wide variety of students who have just finished calculus. The elevation of rigor above all other mathematical virtues gives a skewed view of what mathematics is. My experience with other texts from this genre tells me that the danger of alienating students is very real.

I prefer a rigor-and-survey transition course with twin goals. The first goal is to introduce the students to rigor. The second goal is to give students a better general feel for modern mathematics. Both goals, not just the first, address notable gaps left by their K-13 math education.

The introduction to rigor should be gentle. Students should be given the feeling that they are building on their many years of formal mathematical training, not suddenly entering a foreign land. Rigor should be communicated to them as a corrected version of common sense, not a whole new thing. We should stay entirely away from the vision of mathematics as a giant edifice built on a small number of axioms. Rather we should stay in relative contexts, where students know something about some mathematical objects and try to establish more about these objects using what they already know.

The survey of mathematics should give students a feel for the types of questions asked and answered by modern mathematicians. It should communicate the breadth and excitement of mathematics. It should get students involved in a few landmark theorems and a few open questions. Students should leave the course with some appreciation for what math professors do with their research time!

I have no criticisms of Bloch's book beyond those I have of its entire genre. On the contrary, Proofs and Fundamentals has many strengths. One notable strength is its excellent organization. The book begins by a three-part preface which makes its aims very clear. There are large exercise sets throughout the book. The exercises are well-integrated with the text and vary appropriately from easy to hard. The topics in Part III are quite varied, mostly independent from each other, and truly dependent on Parts I and II. At the end of the book there are useful hints to selected exercises.

Perhaps the book's greatest strength is the author's zeal and skill for helping students write mathematics better. Careful guidance is given throughout the book. Basic issues like not abusing equal signs are treated explicitly. Attention is given to even relatively small issues, like not placing a mathematical symbol directly after a punctuation mark. Throughout the book, theorems are often followed first by informative "scratch work" and only then by proofs. Thus students can see many examples of what they should think, what they should write, and how these are usually not the same.

You may feel as I do that a rigor-and-survey transition course is best. There do not seem to be many texts of this sort, which I suppose suggests that I am advancing a minority viewpoint! I am currently using A Concise Introduction to Pure Mathematics, which I recommend.

On the other hand, you may very well share the opinion of many that transition courses should focus sharply on rigor and basics. In that case, I suggest you seriously consider Proofs and Fundamentals.

David Roberts is an assistant professor of mathematics at University of Minnesota, Morris.